Special Guards in Chromatic Art Gallery

EuroCG 2015, Ljubljana, Slovenia, March 16–18, 2015
Special Guards in Chromatic Art Gallery
Hamid Hoorfar∗
Ali Mohades†
Abstract
We present two new versions of the chromatic art
gallery problem that can improve upper bound of
the required colors pretty well. In our version, we
employ restricted angle guards so that these modern
guards can visit α-degree of their surroundings. If α
is between 0 and 180 degree, we demonstrate that the
strong chromatic guarding number is constant. Then
we use orthogonal 90-degree guards for guarding the
polygons. We prove that the strong chromatic guarding number with orthogonal guards is the logarithmic
order. First, we show that for the special cases of the
orthogonal polygon such as snake polygon, staircase
polygon and mounts polygon, the number of colors is
constant. We decompose the polygon into parts so
that the number of the conflicted parts is logarithmic and every part is snake. Next, we explain the
chromatic art gallery for the orthogonal polygon with
guards that have rectangular visibility. We prove that
the strong chromatic guarding number for orthogonal
polygon with rectangular guards is logarithmic order,
too. We use a partitioning for orthogonal polygon
such that every part is a mount, then we show that
a tight bound for strong chromatic number with rectangular guards is θ(log n)
1
Introduction
New approach of the art gallery problem was raised
with Erickson and LaValle [2], which maximized the
compatible guards so that for two guards whom their
intersection of visibility polygons is not empty must
be spent a new color as cost. In the other words, the
chromatic art gallery find the minimum number of colors that always sufficient and sometimes necessary for
guarding the entire polygon. It is called the chromatic
guarding number. Let χG (P ) denotes the chromatic
guarding number of polygon P. We extend this notation so that χα
G (P ) denotes the chromatic guarding
number with α-degree guards, and χrec
G (P ) denote the
chromatic guarding number with rectangular guards,
properly. The motivation of offering this problem was
in robot controlling with wireless navigators whom we
∗ Laboratory
of Algorithm and Computational Geometry,
Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Iran.
† Laboratory of Algorithm and Computational Geometry,
Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Iran.
can set the angle of their ranges. In many cases, these
navigators have the 90-degree range, and they are orthogonal corresponding to the environment, because
of it, we introduce the chromatic art gallery with orthogonal 90-degree guards in the orthogonal polygons.
Erickson and LaValle showed that for a spiral polygon, the chromatic guarding number is at most 2 and
for a staircase polygon is at most 3, then they showed
that for every positive number k, there exists a polygon with 4k vertices such thatχG (Pk ) ≥ k. Also, they
showed that for every odd numberk there is an orthogonal polygon with 4k 2 + 10k + 10 vertices such
that χG (Pk ) ≥ k [2]. We extend these result with
α-degree guards so that χα
G (P ) ≤ 2 and for orthogonal guards χO
(P
)
∈
O(log
n)
wherever χO
G
G (P ) denotes
the chromatic guarding number with the orthogonal
90-degree guards. After that, we extend these result
with rectangular guards so that χrec
G (P ) ∈ θ(log n)
2
Basic Definitions
Let polygon P be a connected simple subset of R2
with ∂P as its boundary. p ∈ P is visible from q ∈ P
if the segment pq is a subset of P . For every point p
in the polygon, V (p) indicates visibility polygon of p
so that V (p) = {q ∈ P s.t. p is visible from q}. The
guard set SS is a finite set of points in the polygon
such that s∈S V (s) = P , every element of S is called
guard [1]. A pair of guards s and t is named incompatible whenever V (s) ∩ V (t) 6= ∅. Let C(s) denotes
the minimum colors necessary for coloring the guard
set S so that every pair of incompatible guards have
different colors. Furthermore, let T (p) denote the set
of all guard sets in P , and χG (P ) = mins∈T (p) C(s)
is called the chromatic guarding number. The chromatic art gallery problem minimizes the chromatic
guarding number rather the guarding number [3].
Let α-guard denotes the guard whom its visual field
is (v, v + α) wherever v is an arbitrary angle and
α ∈ (0, 180], for instance, 90-guard is a guard with
visual field equal to (v, v + 90). In addition, let
O-guard denotes the orthogonal 90-guard. Suppose
χα
G (p) denotes the chromatic guarding number with
the α-guards, and we extend the notations so that
χO
G (p) indicates the chromatic guarding number with
the O-guards as well. Also:
V α (g) = {p ∈ P | q is visible from α − guard g}
V O (g) = {q ∈ P | q is visible from O − guard g}
In the orthogonal polygon, a horizontal (vertical)
48
This is an extended abstract of a presentation given at EuroCG 2015. It has been made public for the benefit of the community and should be considered a
preprint rather than a formally reviewed paper. Thus, this work is expected to appear in a conference with formal proceedings and/or in a journal.
31st European Workshop on Computational Geometry, 2015
Figure 1: An orthogonal polygon that has no v-cut
edge.
Figure 2: A snake polygon
edge eh (ev ) that its two end points are reflex
vertices is called h-cut edge (v-cut edge) [4]. For
any two points p and q in a rectilinear polygon P,
if the aligned rectangle with p and q as opposite
corners lies totally inside P, then p and q are called
rectangularly visible [4]. Also for any point p in
polygon P, V rec (p) denotes the rectangular visibility
area of P, such that:
V rec (p) = {q ∈ P |p is rectangularly visible from q}
Every guard can see around it rectangularly is
represented by the r-guard. Assume guard set S
that
S is arecfinite set of r-guards in the polygon so that
(s) = P then we call that P is r-visible from
s∈S V
Figure 3: 180-degrees guard
S. we extend the use of the notations to r-guards as
(P
)
=
min
C(S)
whenever
S
is
an
r-guard
χrec
S∈T (p)
G
set. And assume these notations:
Definition 2 Polygon P is called mounts polygon if
V ⊥ (p) = {q ∈ P |q is orthogonally visible from point p}
⊥
it
⊥
V (e) = {q ∈ P |q is orthogonally visible from segment e} has at least an edge e ∈ P such that V (e) = P ,
nominate e, base edge.
Observation 1 Every orthogonal polygon that has
no h-cut edge (v-cut edge) is y-monotone (xmonotone) polygon. As the Figure 1 shows.
In the following we define a new special polygon,
we call it, the snake polygon.
Definition 1 A polygon P is called snake polygon if:
1. P is an orthogonal x-monotone (y-monotone).
2. The line segment created by extending an h-cut
edge to the boundaries of ∂P can decompose P into
exactly three subpolygons.
3. Each of its sub polygon must be xy-monotone. see
Figure 2.
As the sample, every staircase polygon is a snake
polygon; similarly, every xy-monotone polygon is a
snake polygon as well. We define two special polygons, mounts polygon and mount polygon as following.
Definition 3 The mounts polygon P is named
mount polygon if P is monotone corresponding to a
line perpendicular to its base edge.
3
A tight bound on the chromatic guarding number of the general polygon with α-guards
Consider a simple polygon P . Select an arbitrary
edge e1 then place a 180-guard g1 on it. Process
V 180 (g1 ) [5], every connected part of its boundary,
which is not belonged ∂P is called window. Continually place 180-guards on the created windows of previous step until the entire polygon is covered, see Figure
3
We demonstrated that in the general polygon with
180-degree guards the chromatic guarding number is
1, i.e.χ180
G (p) = 1. Now, we can replace every 180guard with at most two colors of α-guards. We want
b 180
α c guards in the same color and perhaps one guard
49
EuroCG 2015, Ljubljana, Slovenia, March 16–18, 2015
Figure 4: Replace a 180-degrees guard with α-guards
Figure 6: The staircase polygon and mount polygon
can be guarding with one color of O-guards
If the bottom edge is not seen, then the visibility
polygon of the O-guards in two different parts will
not be incompatible. We place the green O-guards
in the green areas and the blue ones in the blue areas. Therefore, we can cover the entire snake polygon
with at most two colors. So the chromatic guarding
number of the snake polygon is less than or equal 2.
Observation 4 The chromatic guarding number of
the snake polygon with O-guards is at most 2.
5
Figure 5: O-guarding a snake polygon
in the different color. Wherever 180
α is not an integer
such that it is shown in Figure 4. Hence, if 0 < α ≤
180, then χα
G (p) ≤ 2.
4
Some special polygons with O-guards
Suppose that P is a snake polygon, we present a partitioning so that its chromatic guarding number will
be constant. Extend all h-cut edges in the snake polygon, because of this, the polygon decomposes to sub
polygons such that all of them are staircase or mount
polygons. We can guard all these staircase parts with
one color of the O-guard independently, see Figure 5.
The green parts are the mount polygons, and the blue
ones are staircase polygons.
Observation 2 For every staircase polygon P when
the bottom edge is not seen, χO
G (P ) = 1.
Observation 3 For every mount polygon P when
the bottom edge is not seen, χO
G (P ) = 1, see Figure
6.
50
A tight upper bound on the chromatic guarding number of the orthogonal polygon with Oguards
In this section, we present a partitioning for the orthogonal polygon such that every part is a snake polygon. We construct two partitions in two orientations
both horizontally and vertically. Call horizontal (vertical) partitioning h-cutting (v-cutting), then we nominate duality graph of h-cutting (v-cutting), h-tree
(v-tree). Extend all h-cut (v-cut) edges in the polygon until intersect the boundary, start from the lowest
(leftmost) part, the duality node corresponding to it
must be root, then draw a directed edge from the root
to all its neighbors, continue it recursively until the
h-tree (v-tree) is built. See Figure 7.
We modify any paths in the h-tree, so that duality
of them will be snake polygons. For this purpose, we
cut the v-cut edges occur in the path. We know that
these removed parts will cover with guarding in the
other orientation partitioning, certainly. See Figure 8.
Similarly, consider this process for v-tree. If we
find the chromatic guarding number to cover the
h-tree, double of this number will be the chromatic
guarding number for the entire orthogonal polygon.
We know that every modified path in the duality
tree has the chromatic guarding number at most 2
with the condition that the bottom edge is not seen.
31st European Workshop on Computational Geometry, 2015
Theorem 2 The chromatic guarding number of the
orthogonal polygon with O-guards is the logarithmic
order.
Proof. Using Lemma 1, we know that for guarding
the h-cutting, we need O(log n) colors of guards, by
this guarding, some parts of the polygon is not covered, because of the modified paths, that cut in the
cross orientation. Nevertheless, we are guarding these
remained parts during guarding v-cutting, completely.
For it, we pay O(log n) colors of guards as cost. As
a result, the chromatic guarding number for the orthogonal polygon belongs to O(log n).
6
Figure 7: Every path in the duality graph decomposes
to a snake polygon.
An upper bound on the chromatic guarding
number of the general polygon with O-guards
We present an algorithm for the general polygon such
that we remove an orthogonal polygon from it. By
it, all remained parts are spiral polygons or triangles
so that we can guard them with at most two colors
compatibly. Follow this algorithm:
1. Draw a horizontal line from every vertex in the
polygon.
2. Draw a vertical line from every intersection of
lines and polygon.
3. Select rectangles that occurs in the polygon
completely and make an orthogonal polygon.
4. Consider remained parts, we can guard all disjoint
polygon compatibly.
Figure 8: Partitioning of an orthogonal polygon.
Follow this algorithm:
1. Find the path from the root to the leaf in every
remained tree so that with removing it, the tree
decomposes into at least two sub trees which their
size are at most half of the total tree.
2. Guard all paths from previous step with two new
colors.
3. Remove the path from the total tree and remain
the sub tree(s).
4. If all remained sub trees are not empty, then go to
step 1.
Lemma 1 The number of iterations of the algorithm
is O(log n).
Proof. In iteration the size of the problem will be at
most half of the previous step then we can write:
f (n) ≤ f ( n2 ) + 2 ⇒ f (n) ∈ O(log n), which f (n)
means the complexity of problem.
Observation 5 Each remained part is the spiral
polygon so that reflex chain of it, is orthogonal. (Triangle is a special type of the spiral polygon).
Observation 6 The chromatic guarding number for
the spiral polygon with O-guards in which the reflex
chain is not seen is at most 2.
The condition that the reflex chain doesnt must be
seen appears because, the spiral parts can be guarded
independently.
Lemma 3 The chromatic guarding number for the
general polygon with O-guards is O(log n).
Proof. Decompose a general polygon into orthogonal
polygons that its vertices can be O(n2 ), nevertheless
the χO
G (P ) ∈ O(log n) , for remained spiral polygon
χO
(P
) ≤ 2, therefore, for the general polygon P, we
G
have: χO
G (P ) ∈ O(log n)
7
A tight bound on the rectangular chromatic
guarding number of the orthogonal polygon
Observation 7 The rectangular chromatic guarding
number of the mounts polygon belongs to θ(log n).
51
EuroCG 2015, Ljubljana, Slovenia, March 16–18, 2015
Proof. We found that there is an orthogonal polygon
which is needed θ(log n) as its rectangular chromatic
guarding number, and using above method are shown
that θ(log n) is sufficient for all orthogonal polygons,
so the proof is completed.
8
Figure 9: Partitioning the orthogonal polygon to the
mounts polygons.
We found that the rectangular chromatic guarding
number of mounts polygon is the logarithmic order.
Now, we want to extend this result for all orthogonal polygons as well. First, we divide the polygon
into the mounts polygons. Then, the lowest edge of
the polygon, nominate e and process V ⊥ (e). Suppose
the boundary of V ⊥ (e), every connected part of the
boundary which is not part of the boundary of the
total polygon is called window.
Observation 8 The connected window in V ⊥ (e) is
line segment.
Now, find V ⊥ (e) for every obtained window since
the entire polygon is supported. See Figure 9.
We demonstrate that if we can cover the mounts
polygon with log n colors, then we can cover all parts
of the partitioning with 5 log n colors, in other words:
χrec
G (Porthogonal ) ∈ θ(log n).
If a mounts polygon part has the window in its boundary, then it has incompatibility with other parts. The
incompatibility area is a rectangle such that its width
is equal to width of the window. Start from first part,
color it with the first color and color all parts that immediately left of it (call Left-Parts) with the second
color and all parts that immediately right of it (call
Right-Parts) with the third color. So, color all parts
top of Left-Parts and Right-Parts with 4th color and
all parts bottom of Left-Parts and Right-Parts with
5th color. Any parts in the same color want the same
log n colors of r-guards. By this strategy, we can color
the remained parts with the same five colors so that
showed in Figure 9.
Observation 9 The same colored parts are compatible.
Theorem 4 The rectangular chromatic guarding
number that always sufficient and sometimes necessary for covering an orthogonal polygon is O(log n).
52
Conclusion
In this paper, we explained the chromatic art gallery
on the simple polygon with 90-guards that its visibility field of them is one of the (0, 90), (90, 180),
(180, 270) or (270, 360). This type of guards named
O-guard. In many cases in real-world, the guards can
see a limited angle of its around. This motivated that
we define the new version of the chromatic art gallery
problem so that increase the conflict-free guards from
lower bound n4 to upper bound O(log n). Then, we
explained the chromatic art gallery for the orthogonal polygon with a special type of guards are called
r-guards. This type of guards has r-visibility such that
any two points p and q in a rectilinear polygon P are
called r-visible, if the aligned rectangle with p and q
as opposite corners lies totally inside P .
References
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Oxford University Press, Cambridge, 1987, UK.
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Angeles, USA.
[3] L. H. Erickson. S. M. LaValle. A chromatic art
gallery problem. Technical report, 2010, illinois, USA.
[4] S. K. Ghosh. Visibility Algorithms in the plane. Cambridge University Press, 2007, UK.
[5] H. ElGindy. D. Avis. A Linear algorithm for computing the visibility polygon from a point. The journal
of Algorithms, 1987.